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| Mirrors > Home > MPE Home > Th. List > ereldm | Structured version Visualization version GIF version | ||
| Description: Equality of equivalence classes implies equivalence of domain membership. (Contributed by NM, 28-Jan-1996.) (Revised by Mario Carneiro, 12-Aug-2015.) | 
| Ref | Expression | 
|---|---|
| ereldm.1 | ⊢ (𝜑 → 𝑅 Er 𝑋) | 
| ereldm.2 | ⊢ (𝜑 → [𝐴]𝑅 = [𝐵]𝑅) | 
| Ref | Expression | 
|---|---|
| ereldm | ⊢ (𝜑 → (𝐴 ∈ 𝑋 ↔ 𝐵 ∈ 𝑋)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | ereldm.2 | . . . 4 ⊢ (𝜑 → [𝐴]𝑅 = [𝐵]𝑅) | |
| 2 | 1 | neeq1d 3000 | . . 3 ⊢ (𝜑 → ([𝐴]𝑅 ≠ ∅ ↔ [𝐵]𝑅 ≠ ∅)) | 
| 3 | ecdmn0 8794 | . . 3 ⊢ (𝐴 ∈ dom 𝑅 ↔ [𝐴]𝑅 ≠ ∅) | |
| 4 | ecdmn0 8794 | . . 3 ⊢ (𝐵 ∈ dom 𝑅 ↔ [𝐵]𝑅 ≠ ∅) | |
| 5 | 2, 3, 4 | 3bitr4g 314 | . 2 ⊢ (𝜑 → (𝐴 ∈ dom 𝑅 ↔ 𝐵 ∈ dom 𝑅)) | 
| 6 | ereldm.1 | . . . 4 ⊢ (𝜑 → 𝑅 Er 𝑋) | |
| 7 | erdm 8755 | . . . 4 ⊢ (𝑅 Er 𝑋 → dom 𝑅 = 𝑋) | |
| 8 | 6, 7 | syl 17 | . . 3 ⊢ (𝜑 → dom 𝑅 = 𝑋) | 
| 9 | 8 | eleq2d 2827 | . 2 ⊢ (𝜑 → (𝐴 ∈ dom 𝑅 ↔ 𝐴 ∈ 𝑋)) | 
| 10 | 8 | eleq2d 2827 | . 2 ⊢ (𝜑 → (𝐵 ∈ dom 𝑅 ↔ 𝐵 ∈ 𝑋)) | 
| 11 | 5, 9, 10 | 3bitr3d 309 | 1 ⊢ (𝜑 → (𝐴 ∈ 𝑋 ↔ 𝐵 ∈ 𝑋)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 = wceq 1540 ∈ wcel 2108 ≠ wne 2940 ∅c0 4333 dom cdm 5685 Er wer 8742 [cec 8743 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-br 5144 df-opab 5206 df-xp 5691 df-cnv 5693 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-er 8745 df-ec 8747 | 
| This theorem is referenced by: erth 8796 brecop 8850 eceqoveq 8862 | 
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